From Propagators to Correlators

A differential equation for a Green function fixes neither the quantum state nor every boundary condition. A field correlator additionally specifies operators, a state, and an ordering. In a free theory the familiar vacuum time-ordered correlator is proportional to a Feynman inverse, but an occupied state can change the correlator by a homogeneous solution while leaving its source equation unchanged. The general translation belongs to Green Functions from QM to QFT; this page checks that distinction in the scalar and Dirac conventions used here.

Required background. Green Functions from QM to QFT provides the translation map; Klein–Gordon Propagators fixes scalar source signs; From Fock Space to Quantum Fields supplies the scalar operator construction.

Helpful background. Dirac Propagators fixes the matrix-kernel convention; From Spinors to Fermion Fields derives its field anticommutator.

A correlator includes a state and an ordering

Use $\hbar=c=1$, metric $(+---)$, and free massive fields unless stated otherwise. For operators $A,B$ and a specified density operator $\rho$, define

$$C_{AB}(x,y)=\operatorname{Tr}[\rho A(x)B(y)].$$

Changing $\rho$ or reversing the order generally changes the result. A time-ordered product adds step functions and, for odd fermionic fields, an exchange sign. A retarded response uses a causal bracket with a source convention. These are distinct definitions, even when they involve the same wave operator.

An evolution kernel $\langle x|U(t,t')|y\rangle$ instead acts on state amplitudes. The scalar field’s time-ordered correlator is not generally that one-particle kernel and need not satisfy its composition law. Even a classical second-order scalar initial-value problem requires both initial field and initial time derivative.

Scalar vacuum normalization and causal response

Let $L=\Box+m^2$ and

$$W(z)=\langle0|\widehat\phi(x)\widehat\phi(y)|0\rangle =\int d\Pi_p\,e^{-ip\cdot z}, \qquad z=x-y.$$

The vacuum time-ordered function is

$$D_F(z)=\theta(z^0)W(z)+\theta(-z^0)W(-z).$$

The normalization on the scalar propagator owner gives

$$\widetilde D_F(p)=\frac{i}{p^2-m^2+i0}, \qquad LD_F=-i\delta^{(4)}.$$

Thus the inverse normalized to $LG_F=\delta^{(4)}$ is $G_F=iD_F$. One should check the source equation before transferring a formula from a reference that absorbs the factor of $i$ into its definition.

With a classical perturbation $H_{\rm int}(t)=-\int d^3x\,J(t,\mathbf x)\widehat\phi(t,\mathbf x)$, the free retarded linear-response kernel is

$$G_R(z)=i\theta(z^0) \langle[\widehat\phi(x),\widehat\phi(y)]\rangle.$$

The free commutator is the numerical distribution $W(z)-W(-z)$, independent of the chosen state. Consequently $LG_R=\delta^{(4)}$ with retarded support. State independence here is a property of this linear free-field commutator; general interacting susceptibilities can depend on temperature and preparation.

An occupied mode changes the homogeneous part

Consider one regulated real normal mode of energy $E>0$, with

$$q(t)=\frac{ae^{-iEt}+a^\dagger e^{iEt}}{\sqrt{2E}}, \qquad [a,a^\dagger]=1.$$

In its number state $|n\rangle$, only $\langle aa^\dagger\rangle=n+1$ and $\langle a^\dagger a\rangle=n$ contribute to the two-point function. Hence

$$W_n(t)=\langle n|q(t)q(0)|n\rangle =\frac{(n+1)e^{-iEt}+ne^{iEt}}{2E}.$$

The reversed-order expectation has $t$ replaced by $-t$. Their difference is

$$\langle n|[q(t),q(0)]|n\rangle =-\frac{i}{E}\sin(Et),$$

independent of $n$. Their symmetric average is instead

$$\frac12\langle n|\{q(t),q(0)\}|n\rangle =\frac{2n+1}{2E}\cos(Et).$$

Occupation increases the fluctuations while leaving this mode’s free linear response $g_R(t)=\theta(t)\sin(Et)/E$ unchanged.

The time-ordered function becomes

$$D_n(t)=\frac{e^{-iE|t|}}{2E} +\frac{n}{E}\cos(Et).$$

Its second term is homogeneous: $(\partial_t^2+E^2)\cos(Et)=0$. For every $n$,

$$(\partial_t^2+E^2)D_n(t)=-i\delta(t).$$

The derivative jump remains $-i$; the value at zero changes from $1/(2E)$ to $(2n+1)/(2E)$. Therefore the delta-source equation alone cannot identify the vacuum. The vacuum Feynman prescription is one particular state and boundary choice, not a name for every time-ordered solution of that differential equation.

Nor does an occupied number state’s two-point function automatically determine all its higher correlations by vacuum Wick factorization. The free vacuum is Gaussian; a fixed nonzero number state is generally not.

Fermionic ordering and the Dirac source sign

For the free Dirac field, keep spinor indices explicit in the two orderings:

$$\begin{aligned} S^>_{\alpha\beta}(x,y) &=\langle\widehat\psi_\alpha(x) \widehat{\bar\psi}_\beta(y)\rangle,\\ S^<_{\alpha\beta}(x,y) &=\langle\widehat{\bar\psi}_\beta(y) \widehat\psi_\alpha(x)\rangle. \end{aligned}$$

These definitions contain no extra prefactors of $i$. In the vacuum the time-ordered function is

$$S_F(x-y)=\theta(x^0-y^0)S^>(x,y) -\theta(y^0-x^0)S^<(x,y).$$

Its source equation is

$$(i\gamma^\mu\partial_\mu-m)S_F =i\delta^{(4)}I_4.$$

Accordingly the unit-source Dirac Feynman inverse is $K_F=-iS_F$, whereas the scalar one was $G_F=iD_F$. The signs follow from the respective wave operators and equal-time brackets, not a universal rule that every propagator differs from an inverse by the same factor.

For the free fermionic field the sum $S^>+S^<$ is its state-independent anticommutator $\mathcal A$. The retarded Dirac inverse is $K_R=-i\theta(x^0-y^0)\mathcal A$. This odd-field anticommutator does not replace the commutator in the response of an even physical observable to an ordinary bosonic source. State, observable, source coupling, and statistics must be specified together.

Which data answer the physical question?

Question	Data needed beyond a formal differential inverse
How does an initial state evolve?	State space, initial data, and evolution-kernel normalization
What are vacuum or occupied-state fluctuations?	The quantum state and the requested operator ordering
How does an observable respond to a source?	The observable, source coupling, initial state, and retarded prescription
What scattering rate is measured?	Stable external states, connected amplitudes, normalization, flux, and final-state counting

For the last row, the LSZ mechanism connects appropriate correlator poles to amplitudes. It does not make an arbitrary two-point function equal to a cross section. For a physical source-response question, a vacuum in/out functional likewise need not be the expectation-value functional for a prepared initial state; the next source-functional bridge explains that distinction.

Exercises

The same source, a different state. For $n=2$, find $D_n(0)$ and the homogeneous addition to the vacuum two-point function.

Solution

$D_2(0)=5/(2E)$ and $D_2(t)-D_0(t)=2\cos(Et)/E$. The latter has no derivative jump and is annihilated by $\partial_t^2+E^2$.

A nongeneric Wick rule. In the one-quantum state, use $q^2|1\rangle=(3|1\rangle+\sqrt6|3\rangle)/(2E)$ to compute $\langle q^4\rangle$. Compare it with $3\langle q^2\rangle^2$.

Solution

The squared norm of $q^2|1\rangle$ is $15/(4E^2)$. Since $\langle q^2\rangle=3/(2E)$, the Gaussian Wick value would be $27/(4E^2)$. The mismatch shows why a number state’s two-point function does not justify a Gaussian factorization assumption.

Audit an imported convention. A paper defines $\mathscr D=-i\langle T\phi\phi\rangle$. What is its source equation in this page’s scalar wave-operator convention?

Solution

$L\mathscr D=(-i)(-i)\delta^{(4)} =-\delta^{(4)}$. The symbol is legitimate, but it is not the $+\delta$ inverse $G_F$ without another minus sign. Definitions must be translated before formulas are combined.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press (2014). doi:10.1017/9781139540940. Field correlations, propagators, and scattering interpretation.
• Tong, David. Quantum Field Theory. University of Cambridge lecture notes (2006), sections 2.6–2.7 and 5.4–5.5. Scalar fields; Dirac fields. Free vacuum correlators, locality, and ordering signs.